Problem: Find the number of solutions to
\[\sin x = \left( \frac{1}{2} \right)^x\]on the interval $(0,100 \pi).$
The function $y = \sin x$ and $y = \left (\frac{1}{2} \right)^x$ are plotted below.

[asy]
unitsize (1.5 cm);

real funcf (real x) {
  return (2*sin(pi*x));
}

real funcg (real x) {
  return((1/2)^x);
}

draw(graph(funcf,0,4.2),red);
draw(graph(funcg,0,4.2),blue);
draw((0,-2)--(0,2));
draw((0,0)--(4.2,0));

draw((1,-0.1)--(1,0.1));
draw((2,-0.1)--(2,0.1));
draw((3,-0.1)--(3,0.1));
draw((4,-0.1)--(4,0.1));

label("$\pi$", (1,-0.1), S, UnFill);
label("$2 \pi$", (2,-0.1), S, UnFill);
label("$3 \pi$", (3,-0.1), S, UnFill);
label("$4 \pi$", (4,-0.1), S, UnFill);

label("$y = \sin x$", (4.2, funcf(4.2)), E, red);
label("$y = (\frac{1}{2})^x$", (4.2, funcg(4.2)), E, blue);
[/asy]

On each interval of the form $(2 \pi n, 2 \pi n + \pi),$ where $n$ is a nonnegative integer, the two graphs intersect twice.  On each interval of the form $(2 \pi n + \pi, 2 \pi n + 2 \pi),$ the two graphs do not intersect.  Thus, on the interval $(0, 100 \pi),$ the two graphs intersect $\boxed{100}$ times.